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Publikationen der Gruppe

Hybrid numerical-experimental testing is a standard approach for complex dynamical structures that are, on the one hand, not easy to model due to complexity and parameter uncertainty and, on the other hand, too expensive for full-scale experiments. The main idea is to subdivide the structure in a part that can be accurately simulated with numerical methods and an experimental component. The numerical simulation and the experiment are coupled in real-time by a so-called transfer system, which induces a time-delay into the system. In this paper, we study the solvability of the resulting hybrid numerical-experimental system, which is typically described by a set of nonlinear delay differential-algebraic equations, and extend existing results from the literature to this case.

We study linear time-invariant delay differential-algebraic equations (DDAEs). Such equations arise naturally, if a feedback controller is applied to a descriptor system, since the controller requires some time to measure the state and to compute the feedback resulting in the time-delay. We present an existence and uniqueness result for DDAEs within the space of piecewise smooth distributions and an algorithm to determine whether a DDAE is delay-regular.

We propose a new model reduction framework for problems that exhibit transport phenomena. As in the moving finite element method (MFEM), our method employs time-dependent transformation operators and, especially, generalizes MFEM to arbitrary basis functions. The new framework is suitable to obtain a low-dimensional approximation with small errors even in situations where classical model order reduction techniques require much higher dimensions for a similar approximation quality. Analogously to the MFEM framework, the reduced model is designed to minimize the residual, which is also the basis for an a-posteriori error bound. Moreover, since the dependence of the transformation operators on the reduced state is nonlinear, the resulting reduced order model is obtained by projecting the original evolution equation onto a nonlinear manifold. Furthermore, for a special case, we show a connection between our approach and the method of freezing, which is also known as symmetry reduction. Besides the construction of the reduced order model, we also analyze the problem of finding optimal basis functions based on given data of the full order solution. Especially, we show that the corresponding minimization problem has a solution and reduces to the proper orthogonal decomposition of transformed data in a special case. Finally, we demonstrate the effectiveness of our method with several analytical and numerical examples.

We prove first-order convergence of the semi-explicit Euler scheme combined with a finite element discretization in space for elliptic-parabolic problems which are weakly coupled. This setting includes poroelasticity, thermoelasticity, as well as multiple-network models used in medical applications. The semi-explicit approach decouples the system such that each time step requires the solution of two small and well-structured linear systems rather than the solution of one large system. The decoupling improves the computational efficiency without decreasing the convergence rates. The presented convergence proof is based on an interpretation of the scheme as an implicit method applied to a constrained partial differential equation with delay term. Here, the delay time equals the used step size. This connection also allows a deeper understanding of the weak coupling condition, which we accomplish to quantify explicitly.

We present a graph-theoretical approach that can detect which equations of a delay differential-algebraic equation (DDAE) need to be differentiated or shifted to construct a solution of the DDAE. Our approach exploits the observation that differentiation and shifting are very similar from a structural point of view, which allows us to generalize the Pantelides algorithm for differential-algebraic equations to the DDAE setting. The primary tool for the extension is the introduction of equivalence classes in the graph of the DDAE, which also allows us to derive a necessary and sufficient criterion for the termination of the new algorithm.

This paper is devoted to the construction of exponential integrators of first and second order for the time discretization of constrained parabolic systems. For this extend, we combine well-known exponential integrators for unconstrained systems with the solution of certain saddle point problems in order to meet the constraints throughout the integration process. The result is a novel class of semi-explicit time integration schemes. We prove the expected convergence rates and illustrate the performance on two numerical examples including a parabolic equation with nonlinear dynamic boundary conditions.

Kolmogorov $n$-widths and Hankel singular values are two commonly used concepts in model reduction. Here we show that for the special case of linear time-invariant dynamical (LTI) systems, these two concepts are directly connected. More specifically, the greedy search applied to the Hankel operator of an LTI system resembles the minimizing subspace for the Kolmogorov n-width and the Kolmogorov $n$-width of an LTI system equals its $(n+1)st$ Hankel singular value once the subspaces are appropriately defined. We also establish a lower bound for the Kolmorogov $n$-width for parametric LTI systems and illustrate that the method of active subspaces can be viewed as the dual concept to the minimizing subspace for the Kolmogorov $n$-width.

We consider PDE eigenvalue problems as they occur in two-dimensional photonic crystal modeling. If the permittivity of the material is frequency-dependent, then the eigenvalue problem becomes nonlinear. In the lossless case, linearization techniques allow an equivalent reformulation as an extended but linear and Hermitian eigenvalue problem, which satisfies a Garding inequality. For this, known iterative schemes for the matrix case such as the inverse power or the Arnoldi method are extended to the infinite-dimensional case. We prove convergence of the inverse power method on operator level and consider its combination with adaptive mesh refinement, leading to substantial computational speed-ups. For more general photonic crystals, which are described by the Drude-Lorentz model, we propose the direct application of a Newton-type iteration. Assuming some a priori knowledge on the eigenpair of interest, we prove local quadratic convergence of the method. Finally, numerical experiments confirm the theoretical findings of the paper.

We extend the modeling framework of port-Hamiltonian descriptor systems to include under- and over-determined systems and arbitrary differentiable Hamiltonian functions. This structure is associated with a Dirac structure that encloses its energy balance properties. In particular, port-Hamiltonian systems are naturally passive and Lyapunov stable, because the Hamiltonian defines a Lyapunov function. The explicit representation of input and dissipation in the structure make these systems particularly suitable for output feedback control. It is shown that this structure is invariant under a wide class of nonlinear transformations, and that it can be naturally modularized, making it adequate for automated modeling. We investigate then the application of time-discretization schemes to these systems and we show that, under certain assumptions on the Hamiltonian, structure preservation is achieved for some methods. Numerical examples are provided.

We present a framework for constructing a structured realization of a linear time-invariant dynamical system solely from a discrete sampling of an input and output trajectory of the system. We estimate the transfer function of the original model at selected frequencies using a modification of the empirical transfer function estimation that was recently presented in [Peherstorfer, Gugercin, Willcox, SIAM J. Sci. Comput., 39(5):2152–2178, 2017]. Our realization interpolates the transfer function estimates and can be seen as a generalization of the Loewner framework to structured systems. We demonstrate the presented framework by means of a delay example.

In this paper formulas are derived for the analytic center of the solution set of linear matrix inequalities (LMIs) defining passive transfer functions. The algebraic Riccati equations that are usually associated with such systems are related to boundary points of the convex set defined by the solution set of the LMI. It is shown that the analytic center is described by closely related matrix equations, and their properties are analyzed for continuous- and discrete-time systems. Numerical methods are derived to solve these equations via steepest ascent and Newton-like methods. It is also shown that the analytic center has nice robustness properties when it is used to represent passive systems. The results are illustrated by numerical examples.

In this paper we derive a formulation for Model Predictive Control (MPC) of linear time-invariant systems based on H-infinity loop-shaping. The design provides an optimized stability margin for problems that require state estimation. Input and output weights are designed in the frequency domain to satisfy steady-state and transient performance requirements, in lieu of conventional MPC plant model augmentations. The H-infinity loop-shaping synthesis results in an observer-based state feedback structure. Using the linear state feedback law, an inverse optimal control problem is solved to design the MPC cost function, and the H-infinity state estimator is used to initialize the prediction model at each time step. The MPC inherits the closed-loop performance and stability margin of the loop-shaped design when constraints are inactive. We apply the methodology to a multi-zone heat pump system in simulation. The design rejects constant unmeasured disturbances and tracks constant references with zero steady-state error, has good transient performance, provides an excellent stability margin, and enforces input and output constraints.

We consider the problem of finding an optimal data-driven modal decomposition of flows with multiple convection velocities. To this end, we apply the shifted proper orthogonal decomposition (sPOD) which is a recently proposed mode decomposition technique. It overcomes the poor performance of classical methods like the proper orthogonal de- composition (POD) for a class of transport-dominated phenomena with large gradients. This is achieved by identifying the transport directions and velocities and by shifting the modes in space to track the transports. We propose a new algorithm for computing an sPOD which carries out a residual minimization in which the main cost arises from solving a nonlin- ear optimization problem scaling with the snapshot dimension. We apply the algorithm to snapshot data from the simulation of a pulsed detona- tion combuster and observe that very few sPOD modes are sufficient to obtain a good approximation. For the same accuracy, the common POD needs ten times as many modes and, in contrast to the sPOD modes, the POD modes do not reflect the moving front profiles properly.

We present an anti-windup method in which the Hanus conditioning technique is combined with a user-designed projection of the reference onto the set of feasible steady-state points. This hybrid approach allows the designer to define a policy for steady-state reference tracking which is used to define the reference projection in the case that one or more control inputs saturate. The projection is computed only when the reference input changes, and is therefore less computationally taxing when compared to a command governor or model predictive control, for some applications. We demonstrate the method with an H-infinity loop-shaping controller for a multi-zone heat pump system in simulation.

We introduce a novel model order reduction method for large-scale linear switched systems (LSS) where the coefficient matrices are affected by a low-rank switching. The key idea is to replace the LSS by a non-switched system with extended input and output vectors - called the envelope system - which is able to reproduce the dynamical behavior of the original LSS by applying a certain feedback law. The envelope system can be reduced using standard model order reduction schemes and then transformed back to an LSS. Furthermore, we present an upper bound for the output error of the reduced-order LSS and show how to preserve quadratic Lyapunov stability. The approach is tested by means of various numerical examples demonstrating the efficacy of the presented method.

The propagation of primary discontinuities in initial value problems for linear delay differential-algebraic equations (DDAEs) is discussed. Based on the (quasi-) Weierstraß form for regular matrix pencil, a complete characterization of the different propagation types is given and algebraic criteria in terms of the matrices are developed. The analysis, which is based on the method of steps, takes into account all possible inhomogeneities and history functions and thus serves as the worst-case scenario. Moreover, it reveals possible hidden delays in the DDAE. The new classification for DDAEs is compared to existing approaches in the literature and the impact of splicing conditions on the classification is studied.

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model with inertia. The usual formulation in the form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model with inertia as a polynomial port-Hamiltonian system of differential-algebraic equations, with a quadratic Hamiltonian function including a generalized order parameter. This leads to a robust representation of the system with respect to disturbances: it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, and it explicitly displays all possible constraints and allows for robust simulation methods. The model is immersed into a system of model hierarchies that will be helpful for applying adaptive simulations in future works. We illustrate the advantages of the modified modeling approach with analytics and numerical results.

We consider partial differential equations on networks with a small parameter $\varepsilon$, which are hyperbolic for $\varepsilon>0$ and parabolic for $\varepsilon=0$. With a combination of an $\varepsilon$-expansion and Runge-Kutta schemes for constrained systems of parabolic type, we derive a new class of time discretization schemes for hyperbolic systems on networks, which are constrained due to interconnection conditions. For the analysis we consider the coupled system equations as partial differential-algebraic equations based on the variational formulation of the problem. We discuss well-posedness of the resulting systems and estimate the error caused by the $\varepsilon$-expansion.

We consider linear partial differential equations with an additional delay term, which - under spatial discretization - lead to ordinary differential equations with fixed delay of retarded type. This means that the semi-discrete solution gains smoothness over time. For the concept of classical, mild, and weak solutions we analyse whether this effect also takes place in the original system. We show that some systems behave in a neutral way only. As a result, the smoothness of the exact solution remains unchanged instead of gaining smoothness over time.

Metriplectic systems are state space formulations that have become well-known under the acronym GENERIC. In this work we present a GENERIC based state space formulation in an operator setting that encodes a weak-formulation of the field equations describing the dynamics of a homogeneous mixture of compressible heat-conducting Newtonian fluids consisting of reactive constituents. We discuss the mathematical model of the fluid mixture formulated in the framework of continuum thermodynamics. The fluid mixture is considered an open thermodynamic system that moves free of external body forces. As closure relations we use the linear constitutive equations of the phenomenological theory known as Thermodynamics of Irreversible Processes (TIP). The phenomenological coefficients of these linear constitutive equations satisfy the Onsager-Casimir reciprocal relations. We present the state space representation of the fluid mixture, formulated in the extended GENERIC framework for open systems, specified by a symmetric, mixture related dissipation bracket and a mixture related Poisson-bracket for which we prove the Jacobi-identity.

Transport-dominated phenomena provide a challenge for common mode-based model reduction approaches. We present a model reduction method, which is suited for these kinds of systems. It extends the proper orthogonal decomposition (POD) by introducing time-dependent shifts of the snapshot matrix. The approach, called shifted proper orthogonal decomposition (sPOD), features a determination of the multiple transport velocities and a separation of these. One- and two-dimensional test examples reveal the good performance of the sPOD for transport-dominated phenomena and its superiority in comparison to the POD.

In this work we investigate explicit and implicit difference equations and the corresponding infinite time horizon linear-quadratic optimal control problem. We derive conditions for feasibility of the optimal control problem as well as existence and uniqueness of optimal controls under certain weaker assumptions compared to the standard approaches in the literature which are using algebraic Riccati equations. To this end, we introduce and analyse a discrete-time Lur'e equation and a corresponding Kalman–Yakubovich–Popov (KYP) inequality. We show that solvability of the KYP inequality can be characterized via the spectral structure of a certain palindromic matrix pencil. The deflating subspaces of this pencil are finally used to construct solutions of the Lur'e equation. The results of this work are transferred from the continuous-time case. However, many additional technical difficulties arise in this context.

Abstract We present a framework for constructing structured realizations of linear dynamical systems having transfer functions of the form $\widetilde{C}(\sum_{k=1}^K h_k(s)\widetilde{A}_k)^{-1}\widetilde{B}$ where $h_1, h_2, \ldots, h_K$ are prescribed functions that specify the surmised structure of the model. Our construction is data-driven in the sense that an interpolant is derived entirely from measurements of a transfer function. Our approach extends the Loewner realization framework to a more general system structure that includes second-order (and higher) systems as well as systems with internal delays. Numerical examples demonstrate the advantages of this approach.

A general framework for the regularization of constrained PDEs, also called operator differential-algebraic equations (operator DAEs), is presented. The given procedure works for semi-explicit and semi-linear operator DAEs of first order including the Navier-Stokes and other flow equations. The proposed reformulation is consistent, i.e., the solution of the PDE remains untouched. Its main advantage is that it regularizes the operator DAE in the sense that a semi-discretization in space leads to a DAE of lower index. Furthermore, a stability analysis is presented for the linear case, which shows that the regularization provides benefits also for the application of the Rothe method. For this, the influence of perturbations is analyzed for the different formulations. The results are verified by means of a numerical example with an adaptive space discretization.

As a first step towards time-stepping schemes for constrained PDE systems, this paper presents convergence results for the temporal discretization of operator DAEs. We consider linear, semi-explicit systems which includes e.g. the Stokes equations or applications with boundary control. To guarantee unique approximations, we restrict the analysis to algebraically stable Runge-Kutta methods for which the stability functions satisfy $R(\infty)=0$. As expected from the theory of DAEs, the convergence properties of the single variables differ and depend strongly on the assumed smoothness of the data.

A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.

Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and eciency. The existing package Guptri is very elegant but may sometimes be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specic rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.

A wide class of matrix pencils connected with dissipative Hamiltonian descriptor systems is investigated. In particular, the following properties are shown: all eigenvalues are in the closed left half plane, the nonzero finite eigenvalues on the imaginary axis are semisimple, the index is at most two, and there are restrictions for the possible left and right minimal indices. For the case that the eigenvalue zero is not semisimple, a structure-preserving method is presented that perturbs the given system into a Lyapunov stable system.

We propose a greedy interpolation approach to compute the $\mathcal{L}_\infty$-norm of a possibly irrational $\mathcal{L}_\infty$-function. We approximate this function by a sequence of rational $\mathcal{L}_\infty$-functions. For this we use interpolation employing the Loewner matrix framework. In each iteration, the $\mathcal{L}_\infty$-norm of the rational approximation is computed using established methods. Then a new interpolation point is added where the $\mathcal{L}_\infty$-norm of the function approximation is attained. This way, the $\mathcal{L}_\infty$-norm of the approximation converges superlinearly to the L∞-norm of the original function. We illustrate the efficiency of the resulting algorithm for various numerical examples and compare it to state-of-the-art methods.

A first order perturbation theory for eigenvalues of real or complex J-symplectic matrices under structure- preserving perturbations is developed. As main tools structured canonical forms and Lidskii-like formulas for eigenvalues of multiplicative perturbations are used. Explicit formulas, depending only on appropriately normalized left and right eigenvectors, are obtained for the leading terms of asymptotic expansions describing the perturbed eigenvalues. Special attention is given to eigenvalues on the unit circle, especially to the exceptional eigenvalues 1, whose behavior under structure-preserving perturbations is known to differ significantly from the behavior under general perturbations. Several numerical examples are used to illustrate the asymptotic expansions.

The minimization of operation costs for natural gas transport networks is studied. Based on a recently developed model hierarchy ranging from detailed models of instationary partial differential equations with temperature dependence to highly simplified algebraic equations, modeling and discretization error estimates are presented to control the overall error in an optimization method for stationary and isothermal gas flows. The error control is realized by switching to more detailed models or finer discretizations if necessary to guarantee that a prescribed model and discretization error tolerance is satisfied in the end. We prove convergence of the adaptively controlled optimization method and illustrate the new approach with numerical examples.

In view of highly decentralized and diversified power generation concepts, in particular with renewable energies such as wind and solar power, the analysis and control of the stability and the synchronization of power networks is an important topic that requires different levels of modeling detail for different tasks. A frequently used qualitative approach relies on simplified nonlinear network models like the Kuramoto model. Although based on basic physical principles, the usual formulation in form of a system of coupled ordinary differential equations is not always adequate. We present a new energy-based formulation of the Kuramoto model as port-Hamiltonian system of differential-algebraic equations. This leads to a very robust representation of the system with respect to disturbances, it encodes the underlying physics, such as the dissipation inequality or the deviation from synchronicity, directly in the structure of the equations, it explicitly displays all possible constraints and allows for robust simulation methods. Due to its systematic energy based formulation the model class allows easy extension, when further effects have to be considered, higher fidelity is needed for qualitative analysis, or the system needs to be coupled in a robust way to other networks. We demonstrate the advantages of the modified modeling approach with analytic results and numerical experiments.

The dynamics of elastic media, constrained by Dirichlet boundary conditions, can be modeled as an operator DAE of semi-explicit structure. These models include flexible multibody systems as well as applications with boundary control. In order to use adaptive methods in space, we analyse the properties of the Rothe method concerning stability and convergence for this kind of systems. We consider a regularization of the operator DAE and prove the weak convergence of the implicit Euler scheme. Furthermore, we consider perturbations in the semi-discrete systems which correspond to additional errors such as spatial discretization errors.

We consider Lie and Strang splitting for the time integration of constrained partial differential equations with a nonlinear reaction term. Since such systems are known to be sensitive with respect to perturbations, the splitting procedure seems promising as we can treat the nonlinearity separately. This has some computational advantages, since we only have to solve a linear constrained system and a nonlinear ordinary differential equation. However, Strang splitting suffers from order reduction which limits its efficiency. This reduction is caused by the fact that the nonlinear subsystem produces inconsistent initial values for the constrained subsystem. The incorporation of an additional correction term resolves this problem without increasing the computational cost significantly. Numerical examples including a coupled mechanical system illustrate the proved convergence results.

A model hierarchy that is based on the one-dimensional isothermal Euler equations of fluid dynamics is used for the simulation and optimisation of gas flow through a pipeline network. Adaptive refinement strategies have the aim of bringing the simulation error below a prescribed tolerance while keeping the computational costs low. While spatial and temporal stepsize adaptivity is well studied in the literature, model adaptivity is a new field of research. The problem of finding an optimal refinement strategy that combines these three types of adaptivity is a generalisation of the unbounded knapsack problem. A refinement strategy that is currently used in gas flow simulation software is compared to two novel greedy-like strategies. Both a theoretical experiment and a realistic gas flow simulation show that the novel strategies significantly outperform the current refinement strategy with respect to the computational cost incurred.

In this article, we present a state estimation method for an approximately constant volume combustion process. This is an Unscented Kalman Filter used to estimate quantities which are of interest for the combustion process inside the considered combustion tube, i.e., the pressure, velocity and temperature field. This algorithm relies only on a small number of discrete pressure measurements along the combustion tube. The proposed methods are applied in numerical simulations to demonstrate their effectiveness. In a one-dimensional simulator, the flow field in the tube is described by the one-dimensional Euler equations with chemical source terms. To keep the computational effort on a manageable level, the Kalman Filter is built based on a reduced model. Thus, the model reduction for the Euler equations in combination with chemical kinetics is another focus of this contribution.

Abstract We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

We consider the problem of finding an energy-based formulation of the Navier–Stokes equations for reactive flows. These equations occur in various applications, e. g., in combustion engines or chemical reactors. After modeling, discretization, and model reduction, important system properties as the energy conservation are usually lost which may lead to unphysical simulation results. In this paper, we introduce a port-Hamiltonian formulation of the one-dimensional Navier–Stokes equations for reactive flows. The port-Hamiltonian structure is directly associated with an energy balance, which ensures that a temporal change of the total energy is only due to energy flows through the boundary. Furthermore, the boundary ports may be used for control purposes.

In the simulation and optimization of gas flow in a pipeline network, a hierarchy of models is used that employs different formulations of the Euler equations. While the optimization is performed on piecewise linear models, the flow simulation is based on the simulation of one to three dimensional Euler equations including the temperature distributions. To decide which model class in the hierarchy is adequate to achieve a desired accuracy, this paper presents an error and perturbation analysis for a two level model hierarchy including the isothermal Euler equations in semilinear form and the stationary Euler equations in purely algebraic form. The focus of the work is on the effect of data uncertainty, discretization and rounding errors in the numerical simulation of these models and their interaction. Two simple discretization schemes for the semilinear model are compared with respect to their conditioning and temporal stepsizes are determined for which a well-conditioned problem is obtained. The results are based on new componentwise relative condition numbers for the solution of nonlinear systems of equations. Moreover, the model error between the semilinear and the algebraic model is computed, the maximum pipeline length is determined for which the algebraic model can be used safely, and a condition is derived for which the isothermal model is adequate.

We consider mechanical systems where the dynamics are partially constrained to prescribed trajectories. An example for such a system is a building crane with a load and the requirement that the load moves on a certain path. Enforcing this condition directly in form of a servo constraint leads to differential-algebraic equations (DAEs) of arbitrarily high index. Typically, the model equations are of index 5, which already poses high regularity conditions. If we relax the servo constraints and consider the system from an optimal control point of view, the strong regularity conditions vanish, and the solution can be obtained by standard techniques. By means of the well-known n-car example and an overhead crane, the theoretical and expected numerical difficulties of the direct DAE and the alternative modeling approach are illustrated. We show how the formulation of the problem in an optimal control context works and address the solvability of the optimal control system. We discuss that the problematic DAE behavior is still inherent in the optimal control system and show how its evidences depend on the regularization parameters of the optimization.

We study linear semi-explicit stochastic operator differential algebraic equations (DAEs) for which the constraint equation is given in an explicit form. In particular, this includes the Stokes equations arising in fluid dynamics. We combine a white noise polynomial chaos expansion approach to include stochastic perturbations with deterministic regularization techniques. With this, we are able to include Gaussian noise and stochastic convolution terms as perturbations in the differential as well as in the constraint equation. By the application of the polynomial chaos expansion method, we reduce the stochastic operator DAE to an infinite system of deterministic operator DAEs for the stochastic coefficients. Since the obtained system is very sensitive to perturbations in the constraint equation, we analyze a regularized version of the system. This then allows to prove the existence and uniqueness of the solution of the initial stochastic operator DAE in a certain weighted space of stochastic processes.

We are concerned with the computation of the $\mathcal{L}_\infty$-norm for an $\mathcal{L}_\infty$-function of the form $H(s) = C(s) D(s)^{-1} B(s)$, where the middle factor is the inverse of a meromorphic matrix-valued function, and $C(s),\, B(s)$ are meromorphic functions mapping to short-and-fat and tall-and-skinny matrices, respectively. For instance, transfer functions of descriptor systems and delay systems fall into this family. We focus on the case where the middle factor is large scale. We propose a subspace projection method to obtain approximations of the function $H$ where the middle factor is of much smaller dimension. The $\mathcal{L}_\infty$-norms are computed for the resulting reduced functions, then the subspaces are refined by means of the optimal points on the imaginary axis where the $\mathcal{L}_\infty$-norm of the reduced function is attained. The subspace method is designed so that certain Hermite interpolation properties hold between the largest singular values of the original and reduced functions. This leads to a locally superlinearly convergent algorithm with respect to the subspace dimension, which we prove and illustrate on various numerical examples.

Eigenvalue and eigenpair backward errors are computed for matrix pencils arising in optimal control. In particular, formulas for backward errors are developed that are obtained under block-structure-preserving and symmetry-structure-preserving perturbations. It is shown that these eigenvalue and eigenpair backward errors are sometimes significantly larger than the corresponding backward errors that are obtained under perturbations that ignore the special structure of the pencil.

Low rank perturbations of right eigenvalues of quaternion matrices are considered. For real and complex matrices it is well known that under a generic rank-k perturbation the k largest Jordan blocks of a given eigenvalue will disappear while additional smaller Jordan blocks will remain. In this paper, it is shown that the same is true for real eigenvalues of quaternion matrices, but for complex nonreal eigenvalues the situation is different: not only the largest $k$, but the largest 2k Jordan blocks of a given eigenvalue will disappear under generic quaternion perturbations of rank k. Special emphasis is also given to Hermitian and skew-Hermitian quaternion matrices and generic low rank perturbations that are structure-preserving.

We present a data-driven realization for systems with delay, which generalizes the Loewner framework. The realization is obtained with low computational cost directly from measured data of the transfer function. The internal delay is estimated by solving a least-square optimization over some sample data. Our approach is validated by several examples, which indicate the need for preserving the delay structure in the reduced model.

The present work deals with the inverse dynamics simulation of underactuated mechanical systems relying on servo constraints. The servo-constraint problem of discrete mechanical systems is governed by differential–algebraic equations (DAEs) with high index. We propose a new index reduction approach, which makes possible the stable numerical integration of the DAEs. The new method is developed in the framework of a specific crane formulation and facilitates a reduction from index five to index three and even to index one. Particular attention is placed on the special case in which the reduced index-1 formulation is purely algebraic. In this case the system at hand can be classified as differentially flat system. Both redundant coordinates and minimal coordinates can be employed within the newly developed approach. The success of the proposed method is demonstrated with two representative numerical examples.

Automated modeling of multi-physics dynamical systems often results in large-scale high-index differential-algebraic equations (DAEs). Since direct numerical simulation of such systems leads to instabilities and possibly non-convergence of numerical methods, a regularization or remodeling is required. In many simulation environments, a structural analysis based on the sparsity pattern of the system is used to determine the index and an index-reduced system model. Here, usually the Pantelides algorithm in combination with the Dummy Derivative Method is used. We present a new approach for the regularization of DAEs that is based on the Signature method.

In this master's thesis we adapt recent results on optimal control of continuous-time linear differential-algebraic equations to the discrete-time case of implicit difference equations. First, we adapt equivalent characterizations of solvability of the so-called Kalman- Yakubovich-Popov inequality for differential-algebraic equations to the case of implicit difference equations. That is, we relate the solvability of a certain matrix inequality to the positivity of the Popov function on the unit circle. An essential difference between the continuous-time and the discrete-time linear-quadratic optimal control problem is due to different structures occurring during the analysis in the form of even or palindromic matrix pencils, respectively. Therefore, with the help of certain structured Kronecker canonical forms, we adapt characterizations of inertia of even matrix pencils to palindromic matrix pencils. To this end, we first introduce a suitable notion of inertia for palindromic matrix pencils. These results are used – analogously to the continuous-time case – to characterize solvability of Lur'e equations equivalently by the existence of certain deflating subspaces of the palindromic matrix pencil. Then we use these findings to describe feasibility and the structure of solutions of the linear-quadratic control problem with infinite time horizon. Finally, these results are illustrated by means of an example.

In this report we introduce a Matlab toolbox for the regularization of descriptor systems. We apply it, in particular, for systems resulting from the generalized realization procedure of [16], which generates, via rational interpolation techniques, a linear descriptor system from interpolation data. The resulting system needs to be regularized to make it feasible for the use in simulation, optimization, and control. This process is called regularization.

In this paper a moment-matching procedure for descriptor systems is presented. Based on a time-domain notion of moments, parametrized families of reduced order models matching the moments of the original system are derived. An example of exploiting the flexibility of choosing the free matrix parameters is demonstrated by achieving two-sided moment matching, i. e., obtaining reduced order models of order $\nu$ matching $2\nu$ moments. In this context, a connection with the Loewner framework is shown which is a special case of the framework presented in this paper.

In the simulation of flows, the correct treatment of the pressure variable is the key to stable time-integration schemes. This paper contributes a new approach based on the theory of differential-algebraic equations. Motivated by the index reduction technique of minimal extension, a remodelling of the flow equations is proposed. It is shown how this reformulation can be realized for standard finite elements via a decomposition of the discrete spaces and that it ensures stable and accurate approximations. The presented decomposition preserves sparsity and does not call on variable transformations which might change the meaning of the variables. Since the method is eventually an index reduction, high index effects leading to instabilities are eliminated.

The presented work contains both a theoretical and a statistical error analysis for the Euler equations in purely algebraic form, also called the Weymouth equations or the temperature dependent algebraic model. These equations are obtained by performing several simplifications of the full Euler equations, which model the gas flow through a pipeline. The statistical analysis is performed using both a Monte Carlo Simulation and the Univariate Reduced Quadrature Method and is used to illustrate and confirm the obtained theoretical results.

In this paper we introduce a new matrix nearness problem that is intended to generalize the distance to instability. Due to its applicability in analyzing the robustness of eigenvalues with respect to the arbitrary localization sets (domains) in the complex plane, we call it the distance to delocalization. For the open left half-plane or the unit disk, the distance to the nearest unstable matrix is obtained as a special case. Following the theoretical framework of Hermitian functions and the Lyapunov-type localization approach, we present new Newton-type algorithms for the distance to delocalization: first using an explicit computation of the desired singular values (eD2D), and then using an implicit computation (iD2D). For both algorithms, we introduce a special stabilization technique of the Newton steps and, for a certain class of the localization domains, we provide an additional globality test. Since our investigations are motivated by several practical applications, we illustrate our approach on some of them. Furthermore, in the special case when the distance to delocalization becomes the distance to the continuous time instability, we validate our algorithms against the state-of-the-art computational methods.

Numerous research projects in automotive engineering focus on the industrialization of the thermoelectric generator. The development and the implementation of thermoelectric systems into the vehicle environment are commonly supported by virtual designing activities. In this paper a customized simulation architecture [1] for thermoelectric generators, developed in research project TEG2020, is used to compare two very different TEG concepts: a plate and a tube bundle geometry. The installation space serves as a restriction and therefore is the same for both systems. There are slight advantages in maximum power output, nominal back pressure and overall weight of the tube bundle geometry compared to the lightweight and cost-effective planar thermoelectric generator presented in [1] and [2]. The efficiency of both systems is nearly the same. However there are some disadvantages of the tube bundle geometry in required mass of thermoelectric material, efficiency based on the mass of thermoelectric material, complexity and necessary cooling load.

Recently in [Gu (2011), Comput. Des. Integr. Circuits Syst., 30(9):1207–1320] a procedure was presented that allows to reformulate nonlinear ordinary differential equations in a way that all the nonlinearities become polynomial on the cost of increasing the dimension of the system. We generalize this procedure (called `polynomialization') to systems of differential-algebraic equations (DAEs). In particular, we show that if the original nonlinear DAE is regular and strangeness-free (i. e., it has differentiation index one) then this property is preserved by the polynomial representation. For systems which are not strangeness-free, i. e., where the solution depends on derivatives of the coefficients and inhomogeneities, we also show that the index is preserved for arbitrary strangeness index. However, to avoid ill-conditioning in the representation one should first perform an index reduction on the nonlinear system and then construct the polynomial representations. Although the analytical properties of the polynomial reformulation are very appealing, care has to be given to the numerical integration of the reformulated system due to additional errors. We illustrate our findings with several examples.

This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second order initial-boundary value problems. For the semi-discretization in space, a finite element scheme is presented which satisfies a discrete stability condition. Because of the saddle point structure of the underlying PDE, the resulting system is a DAE of index 3.

We present a model describing simultaneous heat and mass transfer of an absorbing or desorbing laminar liquid film flowing over a vertical isothermal plate. We start with a formulation which is comparable to established models by using simplifying assumptions such as homogeneous velocity and constant film thickness. In contrast to those, we allow for effects like change in properties and differential heat of solution within the bulk of the film. Additionally, enthalpy transport due to interdiffusion is accounted for. The impact of the considered effects are discussed and compared. The numerical solution is obtained by utilising a Newton–Raphson scheme to solve the finite difference formulation of the governing equations. Since the temperature gradients adjacent to wall and phase boundary are expected to be large, we discretise the equations on an irregular grid. The results of the model agree very well with established analytical models. It is found that the influence of released differential heat of solution within the bulk is relatively small. However, the impact on the temperature distribution is in the same order of magnitude as the one of a change in properties. Moreover, when comparing desorption with absorption under equivalent conditions, the mass transfer rate during absorption is higher than during desorption.

R. Kühn, O. Koeppen, P. Schulze, D. Jänsch, M. Pohle, J. Kitte, and J. JägerComparison of simulated and measured data of a thermoelectric generator (TEG) developed in research project TEG 2020 and its hardware in the loop (HIL) resultsIn 4th Conference on Thermoelectrics at IAV, 2014.[bibtex]

@INPROCEEDINGS{KuhKSJPKJ14, title = {Comparison of simulated and measured data of a thermoelectric generator ({TEG}) developed in research project {TEG} 2020 and its hardware in the loop ({HIL}) results}, year = {2014}, booktitle = {4th Conference on Thermoelectrics at IAV}, address = {Berlin, Germany}, author = {K{\"u}hn, R. and Koeppen, O. and Schulze, P. and J{\"a}nsch, D. and Pohle, M. and Kitte, J. and J{\"a}ger, J.},}

In space semi-discretized equations of elastodynamics with weakly enforced Dirichlet boundary conditions lead to differential algebraic equations (DAE) of index 3. We rewrite the continuous model as operator DAE and present an index reduction technique on operator level. This means that a semi-discretization leads directly to an index-1 system. We present existence results for the operator DAE with nonlinear damping term and show that the reformulated operator DAE is equivalent to the original equations of elastodynamics. Furthermore, we show that index reduction and semi-discretization in space commute if the discretization schemes are chosen in an appropriate way.

The $P_1$-nonconforming finite element is introduced for arbitrary triangulations into quadrilaterals and triangles of multiple connected Lipschitz domains. An explicit a priori analysis for the combination of the Park–Sheen and the Crouzeix–Raviart nonconforming finite element methods is given for second-order elliptic PDEs with inhomogeneous Dirichlet boundary conditions.

P. Kunkel and V. MehrmannNecessary and sufficient conditions in the optimal control for general nonlinear differential-algebraic equationsMATHEON, DFG Research Center Mathematics for Key Technologies in Berlin, Preprint, 355, 2006.[bibtex]